Maker-Breaker on Galton-Watson trees

TL;DR

This paper analyzes a Maker-Breaker game on Galton-Watson trees, deriving the probability of Breaker winning using fixed point equations across different information scenarios, offering new insights into combinatorial game theory and random structures.

Contribution

It introduces a novel analysis of Maker-Breaker games on random trees, especially Galton-Watson trees, using fixed point equations to determine winning probabilities under various information regimes.

Findings

Breaker wins with high probability when the tree is sparse.

Maker can secure an infinite path in denser trees.

Winning probabilities depend on the initial edge availability and information level.

Abstract

We consider the following combinatorial two-player game: On the random tree arising from a branching process, each round one player (Breaker) deletes an edge and by that removes the descendant and all its progeny, while the other (Maker) fixates an edge to permanently secure it from deletion. Breaker has won once the tree's root is contained in a finite component, otherwise Maker wins by building an infinite path starting at the root. It will be analyzed both as a positional game (the tree is known to both players at the start) and with more restrictive levels of information (the players essentially explore the tree during the game). Reading the number of available edges for play as a random walk on $\mathbb{Z}$ allows us to derive the winning probability of Breaker via fixed point equations in three natural information regimes. These results provide new insights into combinatorial game theory and random structures, with potential applications to network theory, algorithmic game design and probability theory.